Logic circuits and truth tables
Translate among a problem statement, a Boolean expression, a truth table and a circuit without simplifying the given logic.
Section Component 1 · Computer Systems and Logic
Mapped to the Cambridge IGCSE Computer Science 0265 syllabus for exams in 2029. These original explanations and self-checks are independent study support, not Cambridge papers, official marks or grades. Practical programming is taught with worked code and trace tasks; an executable Python workspace and timed digital-exam simulation are not yet available.
Before you begin
- Use the 2029 syllabus for this course. Cambridge specifies Python 3.10 or higher for programming and digital exams for both components. The syllabus currently identifies this route for 2029 only; confirm later-year availability with Cambridge and your school.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Before logic circuits and truth tables, which statement is the safest starting point?
What you will learn
- Construct a circuit from a problem, expression or truth table
- Complete a truth table from each representation
- Write an expression from a circuit or truth table
Keep each representation linked
A logic circuit is a diagram of gates connected to represent a Boolean expression. Start by identifying each input and the required output, then apply the stated gate relationship directly. The syllabus limits circuits to three inputs and one output, and says not to simplify the given statement.
A truth table enumerates all input combinations. With n inputs there are 2ⁿ rows. Add intermediate columns for gate outputs so the final output can be checked against the full circuit.
Translate in a controlled order
For a circuit-to-expression translation, read from inputs toward the output and place brackets around intermediate results. For an expression-to-circuit translation, build the innermost gate first and feed its output into the next gate.
For a truth table, look for the rows where the output is 1. Write a term for each such row using the input or its negation, then combine the terms with OR. Keep the expression faithful to the table; simplification is not required for the circuit task.
Pause and explain
Which habit best checks your understanding of logic circuits and truth tables?
Worked example
Complete the output for Y = (A AND B) OR C when A=1, B=0 and C=1.
Show the worked solution
- Evaluate the bracket first: 1 AND 0 = 0.
- Use that result with C: 0 OR 1.
- The final output is 1.
Answer Y = 1
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Complete Y = NOT A AND B when A=0 and B=1.
Give me a hint
Apply NOT to A before the AND operation.
Compare my reasoning
- NOT 0 = 1.
- The second input is 1.
- 1 AND 1 = 1.
Y = 1
Look for these in your work
- NOT evaluated
- AND evaluated
- Output stated
Write a Boolean expression for a warning that is on when the temperature is high and either the door is open or the fan has failed.
Give me a hint
Translate the grouped alternatives first, then combine with temperature.
Compare my reasoning
- Let H be high temperature, D be door open, and F be fan failure.
- The alternative trigger is D OR F.
- The warning needs H and that trigger.
H AND (D OR F)
Look for these in your work
- Inputs defined
- OR group represented
- AND condition represented
A table has inputs A, B and output Y. Y is 1 only for rows 00 and 11. Write an expression and explain how it describes the table.
Give me a hint
Identify whether the output is 1 when inputs match or differ.
Compare my reasoning
- Rows 00 and 11 have matching input values.
- XOR is 1 when inputs differ, so its complement is 1 when they match.
- Therefore use NOT (A XOR B), equivalently XNOR if that symbol is available.
Y = NOT (A XOR B)
Look for these in your work
- Rows grouped by output
- Gate behaviour identified
- Expression matches both rows
Common mistakes
- Forgetting that AND is evaluated before a separately grouped OR.
- Omitting an input combination in a truth table.
- Simplifying a circuit when asked to implement the expression as given.
What can you explain now?
How many rows are needed for a complete truth table with three binary inputs?
Compare with the explanation
8
There are 2³ possible input combinations.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
What is a reliable order for translating between a problem, expression, circuit and truth table?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.